Q&A: practice questions with answers

Concept: If the sum of two numbers is known, other number $=\text{sum}-\text{given number}$.

1. Other $=-4-\left(\frac{-13}{25}\right)=-4+\frac{13}{25}$
2. $-4=\frac{-100}{25}$, so other $=\frac{-100+13}{25}=\frac{-87}{25}$

Trick: Check: $\frac{-87}{25}+\frac{-13}{25}=\frac{-100}{25}=-4$ ✓

Answer: (A) $\frac{-87}{25}$

Concept: $V=\frac{4}{3}\pi r^3$, so volume changes as the cube of the radius.

1. New radius $=0.9r$
2. New volume $=(0.9{)}^{3}V=0.729V$
3. Decrease $=1-0.729=0.271=27.1\%$

Trick: For a change of $x\%$ in all three dimensions: $\text{net}=3x+\frac{3x^2}{100}+\frac{x^3}{{100}^2}$ with $x=-10$: $-30+3-0.1=-27.1\%$

Answer: (D) 27.1%

Concept: Father's age $=3\times$ son's age.

1. Son $=15$ years, father $=3\times 15=45$ years
2. Sum $=45+15=60$ years

Trick: Sum $=(3+1)\times \text{son}=4\times 15=60$

Answer: (C) 60 years

Concept: Distance lost due to stoppages per hour = difference of speeds; time stopped $=\frac{\text{lost distance}}{\text{speed without stoppage}}$.

1. Lost distance per hour $=45-36=9$ km
2. Time $=\frac{9}{45}$ h $=\frac{9}{45}\times 60=12$ min

Trick: Stoppage time $=\frac{\text{difference}}{\text{faster speed}}\times 60$ minutes

Answer: (D) 12 min

Concept: Read the values of Company X (blue line) for the required years and subtract.

1. Imports of X in 2019 $=120$ crore
2. Imports of X in 2021 $=80$ crore
3. Difference $=120-80=40$ crore

Trick: Take the label on the correct line only — Company Y (orange) values are a trap here.

Answer: (C) 40 crore

Concept: Shortfall in marks = shortfall in percent of the maximum marks.

1. Shortfall $=975-870=105$ marks
2. $7\%$ of maximum $=105$
3. Maximum $=\frac{105\times 100}{7}=1500$

Trick: Check: pass marks $975=65\%$ of $1500$, Priya $870=58\%$; gap $=7\%$ ✓

Answer: (C) 1500

Concept: $\text{Gain}\%=\frac{\text{Gain}}{\text{CP}}\times 100$

1. A: gain $=480-384=96$, $\frac{96}{384}\times 100=25\%$
2. B: gain $=1672-1254=418$, $\frac{418}{1254}\times 100=33\frac{1}{3}\%$
3. Ratio $=\frac{25}{100/3}=\frac{75}{100}=\frac{3}{4}$

Trick: Use fractions: $\frac{96}{384}=\frac{1}{4}$, $\frac{418}{1254}=\frac{1}{3}$; ratio $=\frac{1/4}{1/3}=\frac{3}{4}$

Answer: (B) $\frac{3}{4}$

Concept: Volume of cylinder $V=\pi r^{2}h$.

1. Let $r=3k$, $h=4k$
2. $\frac{22}{7}\times 9k^2\times 4k=38808\Rightarrow k^3=\frac{38808\times 7}{792}=343$
3. $k=7$, $r=21$ cm, diameter $=2r=42$ cm

Trick: Check options: $d=42\Rightarrow r=21,h=28$; $\frac{22}{7}\times 441\times 28=38808$ ✓

Answer: (D) 42 cm

Concept: $1\text{ cm}=10\text{ mm}$; add mm and cm separately and carry over.

1. mm: $7+6=13$ mm $=1$ cm $3$ mm
2. cm: $5+9+1=15$ cm
3. Total $=15$ cm $3$ mm

Trick: Convert to mm: $57+96=153$ mm $=15.3$ cm

Answer: (D) 15 cm 3 mm

Concept: $0.\overline{abc}=\frac{abc}{999}$ (as many 9s as repeating digits).

1. Let $x=0.512512\dots$
2. $1000x=512.512\dots$
3. $999x=512\Rightarrow x=\frac{512}{999}$

Trick: Pure recurring: repeating block over the same number of 9s.

Answer: (A) $\frac{512}{999}$

Concept: If A takes $a$ hours and B takes $a+6$ hours, then $\frac{1}{a}+\frac{1}{a+6}=\frac{1}{4}$.

1. $4(2a+6)=a(a+6)\Rightarrow a^2-2a-24=0$
2. $(a-6)(a+4)=0\Rightarrow a=6$ hours
3. Check: $\frac{1}{6}+\frac{1}{12}=\frac{3}{12}=\frac{1}{4}$ ✓

Trick: Try options: $a=6$, $b=12$; together $\frac{6\times 12}{6+12}=4$ ✓

Answer: (B) 6 hours

Concept: Volume $=l\times b\times h$, so factorise the expression into three factors.

1. $3x^2-27=3(x^2-9)$
2. $x^2-9=(x-3)(x+3)$
3. So dimensions: $3,(x-3),(x+3)$

Yaad rakhein: $a^2-b^2=(a-b)(a+b)$ after taking out the common factor.

Answer: (C) $3,x-3,x+3$

Concept: Fourth proportional of $a,b,c$ is $d=\frac{b\times c}{a}$ (since $a:b=c:d$).

1. $d=\frac{10\times 12}{15}$
2. $d=\frac{120}{15}=8$

Trick: $15:10=12:d\Rightarrow$ ratio $\frac{2}{3}$, so $d=12\times \frac{2}{3}=8$

Answer: (D) 8

Concept: Let the numbers be $7k$ and $5k$.

1. Difference $=7k-5k=2k=360$
2. $k=180$
3. Greater number $=7\times 180=1260$

Trick: $2$ parts $=360\Rightarrow 1$ part $=180$; greater $=7$ parts.

Answer: (A) 1260

Concept: Division and multiplication are done left to right before addition/subtraction.

1. $3÷3=1$, then $1\times 3=3$
2. $3+3-3=3$

Yaad rakhein: Same-level operations ($÷,\times$) go strictly left to right.

Answer: (D) 3

Concept: $\text{SI}=\frac{P\times R\times T}{100}$; difference depends only on difference of rates.

1. Rate difference $=6.5\%-5.5\%=1\%$
2. Difference $=\frac{300000\times 1\times 2}{100}=6000$

Trick: Same P and T: $\text{diff}=\frac{P\times (R_1-R_2)\times T}{100}$

Answer: (C) ₹6,000

Concept: Form the equation $\frac{5}{4}x-\frac{3}{4}x=7$.

1. $\frac{2}{4}x=7\Rightarrow \frac{x}{2}=7$
2. $x=14$

Trick: Difference of fractions $\frac{5}{4}-\frac{3}{4}=\frac{1}{2}$, so number $=2\times 7$

Answer: (B) 14

Concept: Find each pipe's rate in litres per hour, add, and divide the new capacity.

1. A $=\frac{1000}{5}=200$ L/h, B $=\frac{1000}{6}=\frac{500}{3}$ L/h
2. Together $=200+\frac{500}{3}=\frac{1100}{3}$ L/h
3. Time $=\frac{1800}{1100/3}=\frac{5400}{1100}=\frac{54}{11}$ h

Trick: For 1000 L together: $\frac{5\times 6}{11}=\frac{30}{11}$ h; for 1800 L multiply by $1.8$: $\frac{54}{11}$

Answer: (A) $\frac{54}{11}$

Concept: $\text{Amount}=P+\frac{P\times R\times T}{100}$

1. $\text{SI}=\frac{65000\times 7.5\times 6}{100}=29250$
2. Amount $=65000+29250=94250$

Trick: Options ₹$28750$/₹$29250$ are only interest — question asks for the amount.

Answer: (B) ₹94,250

Concept: Exterior angle $={180}^{\circ }-$ interior angle $=$ sum of the two opposite interior angles.

1. $\angle C={180}^{\circ }-{50}^{\circ }-{70}^{\circ }={60}^{\circ }$
2. Exterior $\angle A={180}^{\circ }-{50}^{\circ }={130}^{\circ }$
3. Check: $\angle B+\angle C={70}^{\circ }+{60}^{\circ }={130}^{\circ }$ ✓

Yaad rakhein: Exterior angle at a vertex $={180}^{\circ }-$ that vertex's angle.

Answer: (A) ${130}^{\circ }$